Simplify; express your answer in exponential form. Assume $q\neq 0, p\neq 0$. $\dfrac{{(q^{3}p^{-4})^{5}}}{{(q^{3}p^{5})^{5}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(q^{3}p^{-4})^{5} = (q^{3})^{5}(p^{-4})^{5}}$ On the left, we have ${q^{3}}$ to the exponent ${5}$ . Now ${3 \times 5 = 15}$ , so ${(q^{3})^{5} = q^{15}}$ Apply the ideas above to simplify the equation. $\dfrac{{(q^{3}p^{-4})^{5}}}{{(q^{3}p^{5})^{5}}} = \dfrac{{q^{15}p^{-20}}}{{q^{15}p^{25}}}$ Break up the equation by variable and simplify. $\dfrac{{q^{15}p^{-20}}}{{q^{15}p^{25}}} = \dfrac{{q^{15}}}{{q^{15}}} \cdot \dfrac{{p^{-20}}}{{p^{25}}} = q^{{15} - {15}} \cdot p^{{-20} - {25}} = p^{-45}$